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Math Help - Odd Primes with Modulo

  1. #1
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    Odd Primes with Modulo

    Please help.



    Let p be an odd prime. Given:

    (x+1)^p = x^p + 1^p (mod p)

    holds for any integer x, prove n^p = n(mod p) for all integers n.



    Any help would be greatly appreciated!!
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  2. #2
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    Quote Originally Posted by zachsch View Post
    Please help.



    Let p be an odd prime. Given:

    (x+1)^p = x^p + 1^p (mod p)

    holds for any integer x, prove n^p = n(mod p) for all integers n.



    Any help would be greatly appreciated!!

    Induction on n: for n= 1 is trivial, so suppose it's true for n-1 and then:

    n^p=(n-1+1)^p\buildrel\text{ind. hyp.}\over=(n-1)^p+1^p\!\!\!\pmod p ...take it from here

    Tonio
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  3. #3
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    How would I go about "taking it from here"?
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  4. #4
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    Quote Originally Posted by zachsch View Post
    How would I go about "taking it from here"?

    Well, for that you'll have to think and make a little effort instead of writing back immediately asking for the solution...

    Tonio
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  5. #5
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    Got it! Thanks a million!!!
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  6. #6
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    By the way you said in your hypothesis that p has to be odd, implying that this does not work for p=2. It does in fact work for p=2.

    Also, the proof of the hypothesis is far more interesting than the proof of your statement, which is practically a corollary of the hypothesis.
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